Wednesday, November 30, 2011

I'm still reading Gravity's Rainbow, which is taking forever. It doesn't help that somewhere in the middle, I decided to take a break, and then the library lost the copy I returned! During my break, I took up a variety of much lighter reading, including Harry Potter and the Methods of Rationality.

This well-known fanfiction takes place in an alternate Potter universe where Harry Potter has been brought up by a scientist. When Harry Potter first discovers the world of magic, he does what any reasonable eleven-year-old would do, and applies the scientific method to it. Harry Potter first tests the existence of magic using pre-agreed protocols. He immediately sees that the wizarding economy is inefficient, and plans to exploit it later. He gets Draco to apply the scientific method to his belief that muggle blood is diluting magic. He tries to use the time-turner to solve NP problems.

Exciting right? I thought it was hilarious. It's also great because it doesn't have as much straw vulcan in it as you might expect. The ideas are mostly taken from the website Less Wrong. Harry Potter isn't right all the time, but that's because he is prey to cognitive bias like anyone else, not because he is "too rational".

But it's not merely about delivering lessons about rationality. Nor is it just about parodying the sillier things the Harry Potter universe. It starts out that way, but soon develops its own story, distinct from the original Harry Potter. It has its share of mysteries, dramas, and dynamic characters. The story discusses many themes that even rational people may disagree on. For example, Professor Quirrell's character has been replaced by a very different character, one who becomes Harry Potter's morally ambiguous mentor.

But since I'm giving this a review, I must also express a few complaints. Though I loved many of the things in the later chapters (eg Hermione calls out Dumbledore for his clear bias towards male heroes), I sort of liked the earlier chapters which were funnier and less serious. I felt the "Humanism" chapters were obnoxious. Harry Potter finds some sappy secular-humanistic way to deal with Dementors. This is pretentious and cliched, and my tastes are too ironic for that sort of thing.

Also, it is much longer than it needs to be. It's not even complete yet, and it's over a thousand pages. I felt way too much time was spent on these wizarding battles that Quirrell uses to teach the kids how to fight. It's a lot like Ender's Game where it goes in great depth about the weightless battles in Ender's military school. I don't particularly care for Ender's Game, or for that part of the book. Quidditch may be an objectively terrible game, but at least it only took up a small part of the books!

But on the whole I enjoyed the work, and I would continue reading it once it gets updated. I highly recommend reading the first few chapters, which are fun even if you don't want to really get into it.

Rules of fillomino:
1. Divide the grid into polyominoes, which are connected shapes made from the little squares. For example, tetris pieces are polyominoes made of four squares.
2. Fill each square with a number, representing the number of little squares in that polyomino.
3. No two polyominoes with the same numbers may share an edge.
4. Some of the numbers are given, but some polyominoes may be implied and have no given numbers.

Solvers voted on the puzzles they liked best. Mine got second, which might reflect on the kind of people who voted. Seriously, they clearly had a taste for rather difficult puzzles.

You may email solutions to skepticsplay at gmail dot com. Enjoy your Thanksgiving!

Saturday, November 19, 2011

There's a new paper on arxiv called "The quantum state cannot be interpreted statistically". It has a theorem which proves that, given a few basic assumptions, the quantum state (ie the wavefunction) must be real, rather than a merely statistical object. Nature has an article which mostly just harps on how "seismic" the paper is.

Nature (correction: the article's author, not Nature itself) compares its importance to Bell's Theorem, which is a very important result indeed from 1964. Bell's theorem proved that if there were "hidden variables" underneath the quantum state, then entangled particles must be communicating with each other faster than light. I've explained Bell's theorem in the past.

I felt the news coverage left a lot of unanswered questions. What do they even mean by the "statistical interpretation" of quantum mechanics? Roughly how is it proven? What is the difference between this and Bell's theorem? I found the answers in the arxiv print, and will attempt to summarize them.

What does the "statistical interpretation" mean?

Let's say that we have two ways of flipping a coin. The first method leads to a 50% chance of heads, and a 50% chance of tails. The second method rigs it so the coin always comes up heads. Let's say that I flipped a coin by one of these two methods, and showed you the result. If the coin was heads, then you would not know which method I used.

Now say that I have two ways of preparing an electron. And suppose that you measured the vertical spin component of the electron. If I use the first method, there is a 50% chance the electron is spin up, and 50% chance spin down. If I use the second method, the electron will always be spin up. If I prepared the electron by one of these two methods, and you found that the electron is spin up, you would not know which method I used.

But electron spin is a little trickier than coin flips, because you can measure the spin component in any direction. Suppose you had tried to measure the horizontal spin component, would you always be able to tell which method I used then? The answer is no. But perhaps there is yet another way to measure it?

The authors equate the "statistical interpretation" with the following: Given any two distinct ways to prepare a quantum state, there is a nonzero probability that the result is consistent with either method of preparation. In other words, no matter what kind of measurement we make, there is a chance that we'll get an outcome that doesn't tell us anything.

What's the difference between this theorem and Bell's Theorem?

Bell's theorem requires that you take many measurements and compile statistics of these measurements. Once you are confident enough in your statistics, you can show that the probabilities are incompatible with the "hidden variable" view of quantum mechanics.

This new theorem requires only one measurement. One measurement, and you're done. (Of course, if you have a noisy experiment, you may need to repeat it to build confidence in your result.)

Of course, the new theorem and Bell's theorem also have a slightly different set of assumptions, and slightly different conclusions. But I think the primary difference is that the new theorem requires one measurement, while Bell's theorem requires compiling statistics.

Roughly how is it proven?

As an example, let's take the two methods of preparing an electron that I described above. It turns out that no matter what measurement I make, there is a chance of an outcome that is consistent with either method A or method B.

But we can be tricky. Let's duplicate the machine that prepares the electrons, and assume that these machines are independent of each other. Now there are four methods of preparation:

A and A (ie both machines use method A)

A and B

B and A

B and B

Suppose that there is a chance that the first machine will produce an electron that is consistent with either method A or method B. There is also a chance that the second machine will produce an electron that is consistent with either method A or method B. Therefore, there is a chance that both machines produce electrons which are consistent with any of the four methods.

But it turns out that there is a measurement we can make with four possible outcomes. And each outcome is inconsistent with one of the methods.

Outcome 1: inconsistent with method 1

Outcome 2: inconsistent with method 2

Outcome 3: inconsistent with method 3

Outcome 4: inconsistent with method 4

What is this special measurement? It's not straightforward. In quantum mechanics, we can measure things like position, momentum, and spin. But we can also measure things like helicity, which tells you whether the spin and momentum are in the same direction, without telling you what direction that is. Similarly, we can measure whether the electrons have spin in the same direction or opposite directions. The measurement described in the paper is sort of like that, but more complicated.

The same theorem can be generalized to any two methods of preparing a quantum state. Suppose that one method always produces a spin up electron, and the other produces a spin up electron 99% of the time. All you have to do is have N duplicates of the electron-producing machine (in this case, N=15 suffices), and take a special measurement. No matter the outcome of this measurement is, it is inconsistent with one of the 2^N possible methods of preparation.

The conclusion is that any two distinct quantum states are not just "probably" different, but always different. You just need a tricky measurement to show it.

Friday, November 18, 2011

Among the interests represented on my blog, puzzles are the oldest. Writing puzzles used to be a hobby of mine back in high school, when I'd write and submit puzzles to a website. Naturally, the website was full of expert puzzle solvers, so that explains why I am unable to write a puzzle of reasonable difficulty.

In high school, I had a half-baked philosophy of puzzle-writing. There are essentially two ways to write a puzzle: Question First, or Answer First.

In the Answer First method, first you think up a clever idea. And then you try to design a puzzle such that the clever idea is the answer. For example, a folk remedy for hiccups is to scare someone. So there's a classic riddle based on this idea:

A man walks into a bar and asks for a glass of water. The bartender pulls out a gun, and the man thanks her. What happened?

I've also written Answer First puzzles of my own. "Fast Clock, Slow Clock" is an unambiguous example, as is "Guess the Meaning". You can usually recognize Answer First puzzles by their clever "Aha!" solutions. All riddles are Answer First puzzles.

In the Question First method, first you think of an interesting problem. And then you check to see if there's an interesting solution to it. For example, a recent puzzle, "Tower of Hanoi Variant" is clearly a Question First puzzle. I was inspired by a problem posed in the game of Freecell; I only tried to find a solution after the fact.

The Answer First method requires quite a bit of creative insight to use, but the Question First method has its own difficulties. When you find an interesting question, there is no guarantee that there is a solution, or that the solution is interesting. And a lot of times, you don't want to just think up one interesting question, you want to think up a whole set of interesting questions. And then you have to look at all of those questions, and see which one has the most interesting answer.

For example, in "Ten Rows of Three", I asked solvers to arrange nine dots into ten rows of three. But I could just as easily ask solvers to arrange X dots into Y rows of Z. What values of X, Y, and Z lead to the most interesting puzzle?

So not only do I need to find a solution without any hints, or even a guarantee that a solution exists, as a puzzle-writer I also have to solve a much larger set of puzzles than the puzzle-solver. This is my secret to being good at puzzle-solving. Write lots of puzzles and then you will become very good at solving them.

But I am not sure that the Question First vs Answer First dichotomy applies to all puzzles (that's why I say the philosophy is half-baked). For example, where does Fillomino fit in? Designing one of these puzzles involves filling more and more clues in, while trying to see what deductions you can make from those clues. But often, the clues we fill in are decided by something that the designer wants in the solution. Depending on the puzzle-designer, it could be more Question First or more Answer First.

My inner skeptic wanted to write a comparison between the Question/Answer First methods of puzzle-writing and the experimental/theoretical methods of science. But my inner skeptic's inner skeptic said that this is ridiculous.

Wednesday, November 16, 2011

Hemant Mehta posted some statistics on Catholic attitudes and beliefs in America. The one-sentence summary is that Catholics don't really fall in line with official Catholic Church teachings.* For example, only 21% of Catholics believe that having celibate male-only clergy is an important aspect of Catholicism, and 60% say you can be a good Catholic without adhering to church teachings on birth control.

*Not news

What struck me, were the statistics on transubstantiation.

In case you didn't catch it, 50% are aware that the Catholic Church teaches that transubstantiation is real rather than symbolic.1 But a higher percentage, 63%, believe it is real. The 17% "unknowing believers" are Catholics who don't know the Catholic Church's teachings, but believe them anyway.

1374 ... In the most blessed sacrament of the Eucharist "the body and blood, together with the soul and divinity, of our Lord Jesus Christ and, therefore, the whole Christ is truly, really, and substantially contained." "This presence is called 'real' - by which is not intended to exclude the other types of presence as if they could not be 'real' too, but because it is presence in the fullest sense: that is to say, it is a substantial presence by which Christ, God and man, makes himself wholly and entirely present."

1381 "That in this sacrament are the true Body of Christ and his true Blood is something that 'cannot be apprehended by the senses,' says St. Thomas, 'but only by faith, which relies on divine authority.' ...

Wholly obtuse and yellowed Dessert wine. A mouthful of smoked ham, sassy french onion soup and a modicum of Baby Ruth bar.

Actually, I took these from a Silly Tasting Note Generator, and I'm sure that if you actually drink wine, they look awfully silly. Not that I can tell. Wikipedia helpfully offers a dictionary of wine tasting descriptors, but I think wine tasting is one of those rare things that you can't learn from the internet.

The asexual community is a bit like a wine tasting community, except that they taste different kinds of attraction. While the dictionary of attraction is not as large as the one for wines, I could come up with a dozen just off the top of my head.* And people still struggle to find words to describe their experiences, often resorting to long stories to do so.

I'm not sure why asexuals are such connoisseurs, but it at least makes sense. Imagine you're in a society that drinks wine, but doesn't care about, and hardly seems to recognize the existence of different kinds of wine. And then you have this group of people who can't stand white wine, but some of them enjoy red wine. Since we know there's at least a distinction between red and white wine, maybe there are distinctions between different types of red wine. Wow, let's go investigate!

And then the red-wine-drinkers tell the rest of society, "Look at all these different flavors of red wine we found. I bet there are lots of flavors of white wine too!" The rest of society shrugs unenthusiastically. Sure, wine is complicated, but do we really need to create so many words to describe it? Wine is wine!

I feel sympathy for both sides, the connoisseurs and the "wine is wine" folks. I don't see why they can't coexist peacefully.

Different flavors of attraction are pretty important to me. Aesthetic attraction and limerence are particularly important to me, because that's what I'm most notably missing. Basically, I don't have a sense of "hotness", or "cuteness", or what have you. And I don't get crushes. Thus it seems obvious to me that we must separate out aesthetic attraction and limerence.

But on the other hand, I can't really tell the difference between sexual and romantic attraction. And I don't really know what platonic attraction is. From the perspective of the asexual community, this is a big blind spot! It's like being unable to distinguish between red and white wine, or not knowing what tannin is.*

*I do not know what tannin is.

Why can't I understand the distinction between romantic and sexual attraction? Maybe I just haven't experienced enough, or I haven't done enough introspection. Maybe I have a genetic insensitivity to a particular flavor. Or maybe the words are poorly defined. Maybe they don't describe single flavors but collections of flavors. Or maybe I'm having trouble connecting the words to their meanings. Nobody can hand me a glass-full and tell me that this is what romantic attraction tastes like. Or maybe everyone else is having the same problem connecting words to meanings, so that the words to really mean different things to different people.

Some people complain to me that asexuals make everything too complicated. All I can do is shrug. Some of those concepts are really important to me, because they hit on a key aspect of my experience. Some words are just meaningless to me, and I only keep track of them as words that are meaningful to other people. Surely, if sexuality is complicated, people are allowed to discuss what exactly is complicated about it for them.

Thursday, November 10, 2011

Why do atheists always focus on Christianity? Why not focus on a much more harmful religion, like Islam?
or...Why do atheists always focus on fundamentalist Christianity? Why not focus on more reasonable forms of religion, like mine?

These are some of those questions people ask vocal atheists (in the US). But to any vocal atheist, a handful of answers are immediately obvious. Here are a few...

The atheist movement prioritizes social change over academic debate. The calm and methodical search for truth is a worthy goal, but one also has to admit the importance of actually getting things done. Religion is not just some academic hypothesis, it's something that actually causes people to kick out their children, oppose LGBT rights, and make poor medical decisions. If that is something we want to change, we do not criticize the most compassionate and reasonable religions, we criticize the least reasonable, most harmful, and most popular religions.

Atheists are focused on here and now. This is not simply due to the desire to create change where it will most benefit ourselves. It is also because we have more power to change the here and now. Islam is pretty bad, but has the most impact far away. Local attitudes are easier to change.

It is likely that you are viewing atheists through a filter. Atheists actually do criticize Islam, liberal Christianity, eastern religions, new age religions, as well as non-religion things like alternative medicine. But the criticism that actually gets widely propagated is the stuff that's most relevant to people, or most exciting. For example, I think atheist criticism of Jainism would be vaguely interesting, but not that interesting, because, honestly, I've never met a Jainist in my life. And I think there is popular appeal in seeing the most extreme atheists and fundamentalist Christians just duke it out. If people want it, then people get it.

Atheists write what they know. That means they'll talk about what's in the news they read. It also means they'll talk about their personal experiences. Even when atheists talk about abstract arguments, they're often inspired by some real argument they had with someone, or an article they read. The same is true about most things people write on any subject.

I'm sure readers can come up with other reasons as well.

My advice to people who think atheists are just focusing on straw men: Relax! If a particular criticism of religious beliefs doesn't apply to your beliefs, then it doesn't apply to your beliefs.

Tuesday, November 8, 2011

Hint: The number of disks you can move increases exponentially as you add more rods.

Still give up?

It's relatively easy to simply show the steps when there are only five rods, but I want to generalize. This requires some recursive algorithms. Advanced puzzle solving ahead!

First, let's define a function G(N), which is the tallest tower we can move if there are N empty rods. In the puzzle, I asked for the tallest tower we can move when there are five rods. Four of those rods are empty, so I basically asked for the value of G(4). I'll give it away right now, G(4) = 15.

But what happens if we have a tower with sixteen disks? We can't simply move the top fifteen disks, and ignore the bottom one. The bottom disk gets in the way, and can't be treated like an empty rod. Therefore, I need to define another function F(N), which is the largest stack of disks we can move from an infinitely tall tower to an empty rod when there are N available empty rods.

Procedure 1: Moving F(N) disks from a big tower to an empty rod, when there are N available empty rods
If N=1, simply move one disk and you're done!
Otherwise...
1. Use Procedure 1 to move F(N-1) disks from the big tower to the first empty rod.
2. Use Procedure 1 to move F(N-2) disks from the big tower to the next empty rod.
3. Continue as above until we get to F(1)
4. Move one disk to the last empty rod.
5. Use Procedure 1 in reverse order to move F(1) disks from the second to last rod onto the last rod.
6. Use Procedure 1 in reverse order to move F(2) disks from the third to last rod onto the last rod.
7. Continue as above until we get to F(N-1).

This procedure allows you to move F(N) = 1 + F(1) + F(2) + ... F(N-1) disks. F(N) is given by this recursive function, but we can also figure out an explicit formula. F(N) = 2^(N-1).

Procedure 2: Moving a tower with G(N) disks to another rod, when there are N available empty rods.
1. Use Procedure 1 to move F(N) disks from the big tower to the first empty rod.
2. Use Procedure 1 to move F(N-1) disks from the big tower to the first empty rod.
3. Continue as above until we get to F(1). By now we should have depleted the big tower of disks.
4. Reverse steps 1 through 3, only now we build the tower on the last rod instead of where it was before.

Friday, November 4, 2011

Previously, I analogized the expansion of the universe to a stretching of a rubber band. Light that travels across the universe is like an ant crawling on that rubber band. Whether the ant ever reaches the other side of the rubber band depends on how fast the rubber band is stretching.

Here, I will introduce something called the scale factor. The scale factor is similar in concept to the total length of the rubber band. The tricky part is that there is no "total length" of the universe, because the universe (as far as we know) is infinitely long. But even if the rubber band were infinitely long, we can still imagine it stretching.

Even without referring to the "ends" of the rubber band, we can still see that the ants are getting further apart. Similarly, even though the universe has no "ends", we can still observe galaxies getting farther apart. There are some complications (like if the ants are crawling around while the rubber band is stretching), but let's assume that clever physicists have found a way to correct for this. The scale factor is proportional to the distance between these two ants/galaxies.

I say "proportional" because it doesn't really matter what the distance is exactly; we just care about how that distance changes over time. Is the scale factor increasing as a constant rate? Or is it slowing down? Or is it accelerating? The answer ultimately depends on the theory of General Relativity, which is outside the scope of this post. There are, at least, some results of General Relativity which make intuitive sense (and others that do not make intuitive sense).

If the universe had no energy, we'd expect the scale factor to increase (or decrease) at a constant rate. "Inertia" is the intuitive explanation. As I explained in my previous post, a constant stretching rate implies that light from any galaxy will eventually reach us.

But since the universe has energy in it, that mass pulls itself together with gravity. So we'd expect the scale factor to slow down. But the rate at which the scale factor slows down depends on what kind of energy it is. If that energy comes from matter, the scale factor slows down at a certain rate. If that energy comes from massless particles like light, the scale factor slows down even at a greater rate.* The difference between matter and light is that they dilute at different rates as the universe stretches.

Now, if the energy comes from something that doesn't dilute at all, then the scale factor actually accelerates over time. I realize this is quite counterintuitive, but I'm postponing the explanation indefinitely until I understand it myself! In any case, that's what dark energy is. Dark energy is a transparent form of energy which doesn't get more dilute as the universe grows.

In summary,Universe without energy: Scale factor increases at constant rateUniverse dominated by radiation: Scale factor slows down over timeUniverse dominated by matter: Scale factor slows down (but not by quite as much)Universe dominated by dark energy: Scale factor accelerates

Of course, the universe has a mix of the different kinds of energies. The early universe was dominated by radiation. But the radiation diluted away as the universe expanded, so the later universe was dominated by matter. But now the matter has diluted away, leaving a universe dominated by dark energy. So at first the scale factor slows down, but it eventually speeds up again.

Click to enlarge. This is based on the ΛCDM (Lambda Cold Dark Matter) model, assuming that the energy of the universe is currently 70% dark energy and 30% matter.

Some definitions to understand the graphs:

The comoving distance is simply the distance divided by the scale factor.

The dotted lines represent the paths of objects, provided they are not "crawling around", but just following the expansion of space.

The particle horizon is the answer to the question: how far can we see? Some 380,000 years after the big bang, the universe became transparent. (Update: the particle horizon is not defined from the point when the universe becomes transparent, but from the "beginning".) From that point on, light traveled from far away objects, and eventually it reached us. Simultaneously, those objects were getting further from us. The particle horizon refers to the current distance of those objects whose light from billions of years ago is just reaching us now. Currently, the particle horizon is 46 billion light years away. Therefore, the size of the observable universe is 93 billion light years across!

The hubble sphere is the distance at which the rate of expansion is equal to the speed of light. Beyond the hubble sphere, objects are receding from us faster than the speed of light. It's allowed to be faster than light, because it's an expansion of space, not motion through space. The relativistic speed limit only applies to the relative motion of objects that are near each other.

The light cone is the region of the universe's past that we can presently see. Remember that the further away we look, the further into the past we look. And the further back into the past, the more dense the universe used to be. This is not the same as the particle horizon, which refers to the current position of galaxies whose past we can see, not their past positions.

The event horizon is the largest region that will ever be in the light cone. If the scale factor were increasing at a constant rate (or decelerating), there would be no event horizon. That is, light from every part of the universe from any time would eventually reach us. But since the scale factor is accelerating, there is an event horizon. There are some parts of the universe that we will never see!

I hope this illuminates some of the many confusing aspects of cosmology. As the paper's title suggests, there are a variety of misconceptions about the expanding universe. One of those misconceptions is that the expansion of the universe isn't allowed to exceed the speed of light. This mistake is even made by physicists when speaking to the public! If you want to read more, I recommend a more popular version of this article, which appeared in Scientific American.

Wednesday, November 2, 2011

I dreamt the story of a little girl with pigtails. Much of the details are lost, but the point was largely to show how bitter she was. Also, there is something wrong with her eyes. She's not blind, but her eyes look strangely dark, and no irises are visible.

I remember the end. The end comes in the form of a flashback. She's walking with a little black boy, her friend. She says bitterly to him, "You know, a black person once stole my eyes." The camera slowly shifts so that we see that at this younger age, her eyes still look mostly normal. She goes off, as the little boy stands there, somewhat angry. After a minute, the boy goes running after her, following her into a room where her mother is. Her mother is black. The boy shouts, "A black person didn't steal your eyes... you are black!" Suddenly, the audience understands... she is an albino black girl, and that's why she's so bitter.

Then my dream replays an earlier scene. There was one scene where it was said that the girl has to stay out of daylight at certain hours and undergo some light therapy. A white woman and man appear in this scene to help her, referring to themselves as "Your father and I". But in retrospect, it seems that they were really talking to a boy who was with her; they were the boy's parents.

The reveal seems like such a shameless and nonsensical retcon. For one thing, the girl obviously couldn't be albino since her hair was black (obviously based on Zimmy from Gunnerkrigg Court). One day I would like to write a work of fiction, but it's never gonna happen if my subconscious comes up with such terrible ideas.